University of British Columbia CPSC 314 Computer Graphics May-June Tamara Munzner. Rasterization, Interpolation, Vision/Color

Transcription

1 University of British Columbia CPSC 34 Computer Graphics May-June 5 Tamara Munzner Rasterization, Interpolation, Vision/Color Week, Thu May 9

2 News reminer: extra lab coverage with TAs - Monays, Wenesays for rest of term just for answering questions, no presentations signup sheet for P emo time Friay -5

3 Reaing: Toay FCG Section. Triangles (Barycentric Coorinates) p 4-46 FCG Chap 3 Raster Algorithms, p except 3.8 FCG Chap 7 Human Vision, p FCG Chap 8 Color, p 3-3 until Section 8.9 Tone Mapping

4 FCG Errata p 54 triangle at bottom of figure shouln t have black outline p 63 The test if numbers a [x] an b [y] have the same sign can be implemente as the test ab [xy] >.

5 Reaing: Next Time FCG Chap 8, Surface Shaing, p 4-5 RB Chap Lighting

6 Clarification: Arbitrary Rotation Y Z W V problem: X U given two orthonormal coorinate systems XYZ an UVW fin transformation from XYZ to UVW answer: transformation matrix R whose columns are U,V,W: u x v x w x R = u y v y w y u z v z w z

7 Review: Projective Renering Pipeline object worl viewing OW OCS WCS WV VCS moeling transformation OCS - object/moel coorinate system WCS - worl coorinate system viewing transformation VCS - viewing/camera/eye coorinate system CCS - clipping coorinate system NDCS - normalize evice coorinate system DCS - evice/isplay/screen coorinate system VC projection transformation CN perspective ivie ND viewport transformation clipping CCS normalize evice NDCS evice DCS

8 Review: Camera Motion rotate/translate/scale ifficult to control arbitrary viewing position eye point, gaze/lookat irection, up vector y lookat WCS x view Pref z eye up Peye

9 Review: Worl to View Coorinates translate eye to origin rotate view vector (lookat eye) to w axis rotate aroun w to bring up into vw-plane y z x WCS v u VCS Peye w Pref up view eye lookat = e w e v e u M z y x z y x z y x w w w v v v u u u v w

10 Correction: Moving Camera or Worl? two equivalent operations move camera one way vs. move worl other way example initial OpenGL camera: at origin, looking along -z axis create a unit square parallel to camera at z = - translate in z by 3 possible in two ways camera moves to z = -3 Note OpenGL moels viewing in left-han coorinates camera stays put, but worl moves to -7 resulting image same either way possible ifference: are lights specifie in worl or view coorinates?

11 Correction: Worl vs. Camera Coorinates C c b a = (,) W b = (,) C = (5,3( 5,3) W c = (,) C = (,3) C = (5,5( 5,5) W a C W

12 Review: Graphics Cameras real pinhole camera: image inverte image plane eye point computer graphics camera: convenient equivalent eye point center of projection image plane

13 Review: Basic Perspective Projection similar triangles similar triangles = z y y' z y y = ' z P(x,y,z) P(x,y,z) P( P(x x,y,z ),z ) z = z = y z x x = ' z = ' z y z x / / z z y x / homogeneous homogeneous coors coors

14 Correction: Perspective Projection esire result for a point [x, y, z, ] T projecte onto the view plane: what coul a matrix look like to o this? z z y z y y z x z x x z y y z x x = = = = = = = ', ', ' ', '

15 Correction: Simple Perspective Projection Matrix = / z y x z z y x / / z y z x is homogenize version of where w = z/ z z y x /

16 Review: Orthographic Cameras center of projection at infinity no perspective convergence just throw away z values = z y x z y x p p p

17 Review: Transforming View Volumes perspective view volume orthographic view volume y VCS x x=left y=top y=bottom z=-near x=right z=-far z VCS x=left y x y=bottom y=top x=right z=-near z=-far NDCS y (,,) (-,-,-) x z

18 Review: Ortho to NDC Derivation scale, translate, reflect for new coor sys VCS NDCS z x=left y y=top x=right (-,-,-) y x z (,,) x y=bottom z=-near z=-far right left P' = top bot far near right + left right left top + bot top bot P far + near far near

19 Review: NDC to Viewport Transformation D scaling an translation (,) (w,h) NDCS DCS b (-,-) x y z DCS DCS DCS ( x +) w ( y +) h ( z +) = NDCS = NDCS = NDCS OpenGL (,) glviewport(x,y,a,b); efault: glviewport(,,w,h); x y a

20 Clarification: NV Transformation general formulation translate by x offset, with/ y offset, height/ scale by with/height reflect in y for upper vs. lower left origin FCG inclues aitional translation for pixel centers at (.5,.5) instea of (,) feel free to ignore this

21 Review: Perspective Normalization perspective viewing frustum transforme to cube orthographic renering of cube prouces same image as perspective renering of original frustum

22 Review: Perspective Normalization viewing VCS projection transformation alter w clipping CCS perspective ivision / w normalize evice NDCS istort such that orthographic projection of istorte objects is esire persp projection separate ivision from stanar matrix multiplies clip after warp, before ivie ivision: normalization

23 Review: Coorinate Systems

24 Review: Perspective Derivation x z NDCS y (-,-,-) (,,) x=left x=right y=top y=bottom z=-near z=-far x VCS y z ) ( n f fn n f f b n t b t b t n l r l r l r n

25 Review: Fiel-of-View Formulation FOV in one irection + aspect ratio (w/h) also set near, far x α Frustum -z z=-n z=-f

26 Projection Wrapup

27 planar projections Projection Taxonomy perspective:,,3-point parallel oblique orthographic cabinet cavalier top, front, sie axonometric: isometric imetric trimetric

28 Perspective Projections classifie by vanishing points one-point perspective two-point perspective three-point perspective

29 Parallel Projection projectors are all parallel vs. perspective projectors that converge orthographic: projectors perpenicular to projection plane oblique: projectors not necessarily perpenicular to projection plane Orthographic Oblique

30 Axonometric Projections projectors perpenicular to image plane select axis lengths

31 Oblique Projections projectors oblique to image plane select angle between front an z axis lengths remain constant both have true front view cavalier: istance true cabinet: istance half / z y cavalier α x z y cabinet α x

32 Demos Tuebingen applets from Frank Hanisch AppletInex.html#Transformationen

33 Rasterization

34 Scan Conversion - Rasterization convert continuous renering primitives into iscrete fragments/pixels lines mipoint/bresenham triangles floo fill scanline implicit formulation interpolation

35 Scan Conversion given vertices in DCS, fill in the pixels start with lines

36 y y assume = mx + b = ( y ( x x < x y goals integer coorinates thinnest line with no gaps, slope how can we o this quickly? ) ( x x) x) Basic Line Drawing + y < y x < Line (, begin float x, y, x, y, slope ; x y slope y y x y x for x from y x y, ; y ; x x to o begin PlotPixel ( x, Roun ( y) y y + slope ; en ; en ; ; x, x y ) ) ;

37 Mipoint Algorithm moving horizontally along x irection raw at current y value, or move up vertically to y+? check if mipoint between two possible pixel centers above or below line caniates top pixel: (x+,y+) bottom pixel: (x+, y) mipoint: (x+, y+.5) check if mipoint above or below line below: top pixel above: bottom pixel key iea behin Bresenham [emo]

38 Making It Fast: Reuse Computation mipoint: if f(x+, y+.5) < then y = y+ on previous step evaluate f(x-, y-.5) or f(x-, y+.5) f(x+, y) = f(x,y) + (y -y ) f(x+, y+) = f(x,y) + (y - y ) + (x - x ) y=y = f(x+, y+.5) for (x=x; x <= x; x++) { raw(x,y); if (<) then { y = y + ; = + (x -x) + (y -y) } else { = + (y -y) }

39 Making It Fast: Integer Only mipoint: if f(x+, y+.5) < then y = y+ on previous step evaluate f(x-, y-.5) or f(x-, y+.5) f(x+, y) = f(x,y) + (y -y ) f(x+, y+) = f(x,y) + (y - y ) + (x - x ) y=y = f(x+, y+.5) for (x=x; x <= x; x++) { raw(x,y); if (<) then { y = y + ; = + (x -x) + (y -y) } else { = + (y -y) } y=y = *(y-y)(x+) + (x- x)(y+) + xy - xy for (x=x; x <= x; x++) { raw(x,y); if (<) then { y = y + ; = + (x -x) + (y -y) } else { = + (y -y) }

40 Rasterizing Polygons/Triangles basic surface representation in renering why? lowest common enominator can approximate any surface with arbitrary accuracy all polygons can be broken up into triangles guarantee to be: planar triangles - convex simple to rener can implement in harware

41 Triangulation convex polygons easily triangulate concave polygons present a challenge

42 OpenGL Triangulation simple convex polygons break into triangles, trivial glbegin(gl_polygon)... glen() concave or non-simple polygons break into triangles, more effort glunewtess(), glutesscallback(),...

43 Problem input: close D polygon problem: fill its interior with specifie color on graphics isplay assumptions simple - no self intersections simply connecte solutions floo fill ege walking

44 Floo Fill simple algorithm raw eges of polygon use floo-fill to raw interior P

45 start with see point Floo Fill recursively set all neighbors until bounary is hit

46 raw eges run: FlooFill (Polygon P, int x, int y, Color C ) if not ( OnBounary ( x, y,p) or Colore ( x, y,c )) begin en ; rawbacks? PlotPixel ( x, y,c ); FlooFill (P, x +, y,c ); FlooFill (P, x, y +,C ); FlooFill (P, x, y,c ); FlooFill (P, x, y,c ); Floo Fill

47 Floo Fill Drawbacks pixels visite up to 4 times to check if alreay set nee per-pixel flag inicating if set alreay must clear for every polygon!

48 Scanline Algorithms scanline: a line of pixels in an image set pixels insie polygon bounary along horizontal lines one pixel apart vertically 3 5= P 4

49 General Polygon Rasterization how o we know whether given pixel on scanline is insie or outsie polygon? B D A C E F

50 General Polygon Rasterization iea: use a parity test for each scanline egecnt = ; for each pixel on scanline (l to r) if (olpixel->newpixel crosses ege) egecnt ++; // raw the pixel if egecnt o if (egecnt % ) setpixel(pixel);

51 Making It Fast: Bouning Box smaller set of caniate pixels loop over xmin, xmax an ymin,ymax instea of all x, all y

52 Triangle Rasterization Issues moving slivers share ege orering

53 Triangle Rasterization Issues exactly which pixels shoul be lit? pixels with centers insie triangle eges what about pixels exactly on ege? raw them: orer of triangles matters (it shouln t) on t raw them: gaps possible between triangles nee a consistent (if arbitrary) rule example: raw pixels on left or top ege, but not on right or bottom ege example: check if triangle on same sie of ege as offscreen point

54 Interpolation

55 Interpolation During Scan Conversion rawing pixels in polygon requires interpolating values between vertices z values r,g,b colour components use for Gourau shaing u,v texture coorinates N x, N y, N z surface normals equivalent methos (for triangles) bilinear interpolation barycentric coorinates

56 Bilinear Interpolation interpolate quantity along L an R eges, as a function of y then interpolate quantity as a function of x P P 3 y P L P R P(x,y) P

57 Barycentric Coorinates weighte combination of vertices smooth mixing speeup P compute once per triangle = α P γ α + β + γ = α, β, γ convex combination of points + β P + P3 P 3 for points insie triangle P P (α,β,γ α,β,γ) ) = (,,) β = (α,β,γ) ) = (,,) β =. 5 P β = (α,β,γ α,β,γ) ) = (,,)

58 Deriving Barycentric Coorinates I non-orthogonal coorinate system P 3 is origin P -P 3, P -P 3 are basis vectors P P P = P 3 = α( P + β ( P 3 = ( β γ ) P 3 ) + β ( P P 3 ) + γ ( P + β ( P ) + γ ( P ) + γ ( P ) P 3 ) ) (,,) P (,,) P 3 P P (,,)

59 P P 3 P P L P R P : : ) ( ) ( P P P P P P P P L = = = + + = Deriving Barycentric Coorinates II from bilinear interpolation of point P on scanline

60 Deriving Barycentric Coorinates II similarly b : b : b P P 3 P P L P R P : : ) ( ) ( P b b b P b b b P b b b P b b b P P b b b P P R = = = + + =

61 L P R c c c P c c c P = b : b : b P P 3 P P L P R P : : 3 P P P L = P b b b P b b b P R = c : c : c = 3 P b b b P b b b c c c P P c c c P Deriving Barycentric Coorinates II combining gives

62 Deriving Barycentric Coorinates II thus with can verify barycentric properties 3 3 P a P a P a P + + = c c c b b b c c c c c c b b b c c c + + = = + + = γ β α,,, = + + γ β α γ β α

63 Deriving Barycentric Coorinates III D triangle area α = A β = A γ = A P P P 3 / A / A / A A = + A + A + P A 3 P P (α,β,γ) ) = (,,) P 3 A P A P P A P3 P (α,β,γ α,β,γ) ) = (,,) P (α,β,γ α,β,γ) ) = (,,)

64 Vision/Color

65 Simple Moel of Color simple moel base on RGB triples component-wise multiplication of colors (a,a,a) * (b,b,b) = (a*b, a*b, a*b) why oes this work?

66 elements of color: Basics Of Color

67 physics illumination Basics of Color electromagnetic spectra reflection material properties surface geometry an microgeometry (i.e., polishe versus matte versus brushe) perception physiology an neurophysiology perceptual psychology

68 Electromagnetic Spectrum

69 White Light sun or light bulbs emit all frequencies within the visible range to prouce what we perceive as the "white light"

70 Sunlight Spectrum

71 White Light an Color when white light is incient upon an object, some frequencies are reflecte an some are absorbe by the object combination of frequencies present in the reflecte light that eterminses what we perceive as the color of the object

72 Hue hue (or simply, "color") is ominant wavelength/frequency integration of energy for all visible wavelengths is proportional to intensity of color

73 Saturation or Purity of Light how washe out or how pure the color of the light appears contribution of ominant light vs. other frequencies proucing white light saturation: how far is color from grey pink is less saturate than re, sky blue is less saturate than royal blue

74 Intensity vs. Brightness intensity : measure raiant energy emitte per unit of time, per unit soli angle, an per unit projecte area of the source (relate to the luminance of the source) lightness/brightness : perceive intensity of light nonlinear

75 the retina ros b/w, eges cones color! Physiology of Vision

76 Physiology of Vision center of retina is ensely packe region calle the fovea. cones much enser here than the periphery

77 Foveal Vision hol out your thumb at arm s length X Y G W H C D K A N O S R P Q B E F J L M V T U

78 three types of cones Trichromacy L or R, most sensitive to re light (6 nm) M or G, most sensitive to green light (56 nm) S or B, most sensitive to blue light (43 nm) color blinness results from missing cone type(s)

79 Metamers a given perceptual sensation of color erives from the stimulus of all three cone types ientical perceptions of color can thus be cause by very ifferent spectra

80 Metamer Demo

81 Aaptation, Surrouning Color color perception is also affecte by aaptation (move from sunlight to ark room) surrouning color/intensity: simultaneous contrast effect

82 impact of outlines Bezol Effect

83 Color/Lightness Constancy

84 Color/Lightness Constancy

85 Color/Lightness Constancy

86 Color/Lightness Constancy

87 Color/Lightness Constancy

88 Color/Lightness Constancy

89 Color Constancy automatic white balance from change in illumination vast amount of processing behin the scenes! colorimetry vs. perception

90 Stroop Effect re blue orange purple green

91 Stroop Effect blue green purple re orange interplay between cognition an perception

92 Color Spaces three types of cones suggests color is a 3D quantity. how to efine 3D color space? iea: perceptually base measurement shine given wavelength (λ) on a screen user must control three pure lights proucing three other wavelengths (say R=7nm, G=546nm, an B=436nm) ajust intensity of RGB until colors are ientical this works because of metamers!

93 Negative Lobes exact target match with phosphors not possible some re ha to be ae to target color to permit exact match using knobs on RGB intensity output of CRT equivalently theoretically to removing re from CRT output figure shows that re phosphor must remove some cyan for perfect match CRT phosphors cannot remove cyan, so 5 nm cannot be generate

94 Negative Lobes can t generate all other wavelenths with any set of three positive monochromatic lights! solution: convert to new synthetic coorinate system to make the job easy

95 CIE Color Space CIE efine three imaginary lights X, Y, an Z, any wavelength λ can be matche perceptually by positive combinations Note that: X ~ R Y ~ G Z ~ B

96 Measure vs. CIE Color Spaces measure basis monochromatic lights physical observations negative lobes transforme basis imaginary lights all positive, unit area Y is luminance, no hue X,Z no luminance

97 CIE Gamut an Chromaticity Diagram 3D gamut chromaticity iagram hue only, no intensity

98 RGB Color Space (Color Cube) efine colors with (r, g, b) amounts of re, green, an blue use by OpenGL harware-centric RGB color cube sits within CIE color space subset of perceivable colors scale, rotate, shear cube

99 Device Color Gamuts use CIE chromaticity iagram to compare the gamuts of various evices X, Y, an Z are hypothetical light sources, no evice can prouce entire gamut

100 Gamut Mapping

101 Aitive vs. Subtractive Colors aitive: light monitors, LCDs RGB moel subtractive: pigment printers CMY moel = B G R Y M C

102 HSV Color Space more intuitive color space for people H = Hue S = Saturation V = Value or brightness B or intensity I or lightness L Hue Saturation Value

103 HSI Color Space conversion from RGB not expressible in matrix 3 B G R I + + = I B G R S ) min( + + = [ ] + + = ) )( ( ) ( ) ( ) ( cos B G B R G R B R G R H

104 YIQ Color Space color moel use for color TV Y is luminance (same as CIE) I & Q are color (not same I as HSI!) using Y backwars compatible for B/W TVs conversion from RGB is linear Y R I = G Q..5.3 B I Q green is much lighter than re, an re lighter than blue

105 Luminance vs. Intensity luminance Y of YIQ.99R +.587G +.4B intensity/brightness I/V/B of HSI/HSV/HSB.333R +.333G +.333B

106 Monitors monitors have nonlinear response to input characterize by gamma isplayeintensity = a γ (maxintensity) gamma correction / γ γ isplayeintensity = a (maxintensity) = a (maxintensity)

107 Alpha transparency (r,g,b,α) fraction we can see through c = αc f + (-α)c b compositing

108 Program : Terrain Navigation make colore terrain x gri two triangles per gri cell face color varies ranomly

109 Navigating two flying moes: absolute an relative absolute keyboar keys to increment/ecrement x/y/z position of eye, lookat, up vectors relative mouse rags incremental wrt current camera position forwar/backwar motion roll, pitch, an yaw angles

110 Hints: Viewing on t forget to flip y coorinate from mouse winow system origin upper left OpenGL origin lower left all viewing transformations belong in moelview matrix, not projection matrix project template incorrect with this!

111 Hint: Incremental Motion motion is wrt current camera coors maintaining cumulative angles wrt worl coors woul be ifficult computation in coor system use to raw previous frame is simple OpenGL moelview matrix has the info! but multiplying by new matrix gives p =CIp you want to o p =ICp trick: ump out moelview matrix wipe the stack with glientity apply incremental upate matrix apply current camera coor matrix

112 Demo